3.3 The Unit Circle and Circular Functions
Circular Functions
Unit Circle 
Circular Functions
For any real number s represented by a directed arc on the unit circle.
The Unit Circle
Symmetry Unit circle is symmetric respect to the x-axis, y-axis and origin
(Quadrant I), 
(Quadrant II),
(Quadrant III), 
(Quadrant IV)
In General if 
is in first quadrant –
the angles in other quadrants are:
, 
, 
, 
Specific Example
for 
|
s |
Quadrant of s |
Symmetry Type and Corresponding Point |
Cos s |
Sin s |
|
|
I |
Not applicable; |
|
|
|
|
II |
y-axis; |
|
|
|
|
III |
Origin; |
|
|
|
|
IV |
x-axis; |
|
|
Note: 
Range: x, y are on the unit circle thus
Domains of the Circular Functions
Sine and
Cosine Functions: 
Tangent and Secant Functions: 
Cotangent and Cosecant Functions: 
Finding Values of
Circular Functions
The circular functions of real numbers correspond to the trigonometric functions of angles measured in radians.
Finding Exact
Circular Function Values
Use the unit circle to find the exact values of
sin(
)
cos()
tan()
sin
cos
tan
sec
Approximate Circular Function Values **Radian Mode on Calculator**
Find a calculator approximation for each circular function value:
sin( 3.42)
tan( .8234)
sec( 5.6041)
csc( - 2.7335)
Find s given its Circular Function Value
a) Approximate the value of s in 
if sin s = .3210
b) Find the exact value of s in 
if tan s =
Application
The angle of elevation θ of
the sun in the sky at any latitude L
is calculated with the formula 
where 
corresponds to sunrise
and 
occurs if the sun is
directly overhead. 
(the
Greek letter omega) is the number of
radians that Earth has rotated through since noon, when ω = 0. D
is the declination of the sun, which varies because Earth is tilted on its
axis.
Sitka, Alaska has latitude L = 57.1o. Find the angle of elevation θ of the sun at 3 p.m. on February 29, 2008, where at that time D ≈ -.1425 and
ω ≈ .7854. (Ans: 15.1o)
Linear Measures of Circular Functions
